On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in RN

By Claudianor Oliveira Alves, Giovany M. Figueiredo. 2009

160 citations recorded by the source corpus.

Abstract In this work we study the existence, multiplicity and concentration of positive solutions for the following class of quasilinear problems − ϵ N Δ N u + V ( x ) | u | N − 2 u = f ( u ) in R N , u ( x ) > 0 , ∀ x ∈ R N , where Δ N u is the N-Laplacian operator, ϵ is a positive parameter, N ⩾ 2 , V : R N → R is a continuous function and f : R → R is a C 1 function having critical exponential growth.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/11463726d2016546931fd11440f177768bdc6f38

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In Applied Mathematics

1. Gongbao Li
2. Caifeng Zhang
3. Federica Sani
4. Hongyu Ye
5. Nader Masmoudi
6. Shibo Liu
7. Y. Yang
8. G. Lu

In Computational Theory and Mathematics

1. Caifeng Zhang
2. Federica Sani
3. Gongbao Li
4. Nader Masmoudi
5. Hongyu Ye
6. Shibo Liu
7. Y. Yang
8. Nikolaos S. Papageorgiou

Citations in the corpus

• Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation (2018, methodology): “With the help of the standardmountain-pass procedure, the authors of [3, 5, 8, 16, 32, 38] established the existence of nontrivial weak solutions.”
• Trudinger-Moser Inequalities with the Exact Growth Condition in ℝ N and Applications (2015, background): “The authors of [4, 5, 13, 14, 16, 17, 20, 30], considering a potential V satisfying (1.”
• Existence of positive solutions to quasi-linear elliptic equations with exponential growth in the whole Euclidean space (2011, background): “In fact, in [7, 12, 18, 29], all the contributors only used the above subcritical inequality.”
• On ground states of superlinear p-Laplacian equations in RN (2010, background): “In recent years more and more attention is paid to the quasilinear elliptic equations setting on RN .T o name af ew we mention [1, 2 ,11].”
• Existence of positive solutions to some nonlinear equations on locally finite graphs (2016, unknown): “ly positive solutions. This kind of problems have been extensivelystudied in the Euclideanspace, see for examples Alama-Li [4], Adimurthi [1], Adimurthi-Yadava[2], Adimuthi-Yang [3], Alves-Figueiredo [5], Cao [7], Ruf et al [8, 9], Ding-Ni [10], doO et al [11, 12, 13, 14], Jeanjean [16], Kryszewski-”
• Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $R^3$ (2013, methodology): “nal I and the manifold M are symmetric, hence we may assume that such a ground state solution does not change sign, i.e. ˜u ≥ 0. By using the strong maximum principle and standard arguments, see e.g. [2, 8, 20, 25, 31, 32], we obtain that ˜u(x) > 0 for all x ∈ R3. Therefore, ˜u is a positive grou”
• Existence of positive solutions for nonlinear Kirchhoff type problems in R3 with critical Sobolev exponent (2013, methodology): “First, since p ∈ [2, 4), condition (F ) does not hold, let alone (AR).”